\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  0.0 & + 10.00 x_{1} & +  8.00 x_{2} & + 10.00 x_{3} & +  1.00 x_{4} & -10.00 x_{5} & +  2.00 x_{6} & -6.00 x_{7} & -8.00 x_{8}\\
 x_{10}   &  0.0 & -4.00 x_{1} & +  3.00 x_{2} & +  7.00 x_{3} & -7.00 x_{4} & -1.00 x_{5} & -3.00 x_{6} & +  6.00 x_{7} & +  2.00 x_{8}\\
 x_{11}   &  -3.0 & +  6.00 x_{1} & +  9.00 x_{2} & +  9.00 x_{3} & +  1.00 x_{4} & +  5.00 x_{5} & -2.00 x_{6} & +  3.00 x_{7} & -8.00 x_{8}\\
 x_{12}   &  3.0 & +  9.00 x_{1} & -5.00 x_{2} & -5.00 x_{3} & -4.00 x_{4} & -1.00 x_{5} & -5.00 x_{6} & +  6.00 x_{7} & -7.00 x_{8}\\
 x_{13}   &  2.0 & +  2.00 x_{1} & +  5.00 x_{2} & -7.00 x_{3} & -7.00 x_{4} & -2.00 x_{5} & -4.00 x_{6} &   & +  6.00 x_{8}\\
\hline
z    &  0.0 & -5.00 x_{1} & -4.00 x_{2} & +  4.00 x_{3} & +  3.00 x_{4} & -1.00 x_{5} & -5.00 x_{6} & +  5.00 x_{7} & -4.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -10.00 y_{9} & +  4.00 y_{10} & -6.00 y_{11} & -9.00 y_{12} & -2.00 y_{13}\\
 y_{2}   &  1.0 & -8.00 y_{9} & -3.00 y_{10} & -9.00 y_{11} & +  5.00 y_{12} & -5.00 y_{13}\\
 y_{3}   &  1.0 & -10.00 y_{9} & -7.00 y_{10} & -9.00 y_{11} & +  5.00 y_{12} & +  7.00 y_{13}\\
 y_{4}   &  1.0 & -1.00 y_{9} & +  7.00 y_{10} & -1.00 y_{11} & +  4.00 y_{12} & +  7.00 y_{13}\\
 y_{5}   &  1.0 & + 10.00 y_{9} & +  1.00 y_{10} & -5.00 y_{11} & +  1.00 y_{12} & +  2.00 y_{13}\\
 y_{6}   &  1.0 & -2.00 y_{9} & +  3.00 y_{10} & +  2.00 y_{11} & +  5.00 y_{12} & +  4.00 y_{13}\\
 y_{7}   &  1.0 & +  6.00 y_{9} & -6.00 y_{10} & -3.00 y_{11} & -6.00 y_{12} &   \\
 y_{8}   &  1.0 & +  8.00 y_{9} & -2.00 y_{10} & +  8.00 y_{11} & +  7.00 y_{12} & -6.00 y_{13}\\
\hline
z    &  -0  &    &   & +  3.00 y_{11} & -3.00 y_{12} & -2.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  0.333333333333 & -4.67 y_{9} & +  6.00 y_{10} & +  0.67 y_{2} & -12.33 y_{12} & +  1.33 y_{13}\\
 y_{11}   &  0.111111111111 & -0.89 y_{9} & -0.33 y_{10} & -0.11 y_{2} & +  0.56 y_{12} & -0.56 y_{13}\\
 y_{3}   &  0.0 & -2.00 y_{9} & -4.00 y_{10} & +  1.00 y_{2} &   & + 12.00 y_{13}\\
 y_{4}   &  0.888888888889 & -0.11 y_{9} & +  7.33 y_{10} & +  0.11 y_{2} & +  3.44 y_{12} & +  7.56 y_{13}\\
 y_{5}   &  0.444444444444 & + 14.44 y_{9} & +  2.67 y_{10} & +  0.56 y_{2} & -1.78 y_{12} & +  4.78 y_{13}\\
 y_{6}   &  1.22222222222 & -3.78 y_{9} & +  2.33 y_{10} & -0.22 y_{2} & +  6.11 y_{12} & +  2.89 y_{13}\\
 y_{7}   &  0.666666666667 & +  8.67 y_{9} & -5.00 y_{10} & +  0.33 y_{2} & -7.67 y_{12} & +  1.67 y_{13}\\
 y_{8}   &  1.88888888889 & +  0.89 y_{9} & -4.67 y_{10} & -0.89 y_{2} & + 11.44 y_{12} & -10.44 y_{13}\\
\hline
z    &  0.333333333333 & -2.67 y_{9} & -1.00 y_{10} & -0.33 y_{2} & -1.33 y_{12} & -3.67 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  2.66666666667 & +  4.67 x_{1} & +  0.89 x_{11} & +  2.00 x_{3} & +  0.11 x_{4} & -14.44 x_{5} & +  3.78 x_{6} & -8.67 x_{7} & -0.89 x_{8}\\
 x_{10}   &  1.0 & -6.00 x_{1} & +  0.33 x_{11} & +  4.00 x_{3} & -7.33 x_{4} & -2.67 x_{5} & -2.33 x_{6} & +  5.00 x_{7} & +  4.67 x_{8}\\
 x_{2}   &  0.333333333333 & -0.67 x_{1} & +  0.11 x_{11} & -1.00 x_{3} & -0.11 x_{4} & -0.56 x_{5} & +  0.22 x_{6} & -0.33 x_{7} & +  0.89 x_{8}\\
 x_{12}   &  1.33333333333 & + 12.33 x_{1} & -0.56 x_{11} &   & -3.44 x_{4} & +  1.78 x_{5} & -6.11 x_{6} & +  7.67 x_{7} & -11.44 x_{8}\\
 x_{13}   &  3.66666666667 & -1.33 x_{1} & +  0.56 x_{11} & -12.00 x_{3} & -7.56 x_{4} & -4.78 x_{5} & -2.89 x_{6} & -1.67 x_{7} & + 10.44 x_{8}\\
\hline
z    &  -0.333333333333 & -0.33 x_{1} & -0.11 x_{11} &   & -0.89 x_{4} & -0.44 x_{5} & -1.22 x_{6} & -0.67 x_{7} & -1.89 x_{8}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  2.66666666667 & +  4.67 x_{1} & +  0.89 x_{11} & +  2.00 x_{3} & +  0.11 x_{4} & -14.44 x_{5} & +  3.78 x_{6} & -8.67 x_{7} & -0.89 x_{8}\\
 x_{10}   &  1.0 & -6.00 x_{1} & +  0.33 x_{11} & +  4.00 x_{3} & -7.33 x_{4} & -2.67 x_{5} & -2.33 x_{6} & +  5.00 x_{7} & +  4.67 x_{8}\\
 x_{2}   &  0.333333333333 & -0.67 x_{1} & +  0.11 x_{11} & -1.00 x_{3} & -0.11 x_{4} & -0.56 x_{5} & +  0.22 x_{6} & -0.33 x_{7} & +  0.89 x_{8}\\
 x_{12}   &  1.33333333333 & + 12.33 x_{1} & -0.56 x_{11} &   & -3.44 x_{4} & +  1.78 x_{5} & -6.11 x_{6} & +  7.67 x_{7} & -11.44 x_{8}\\
 x_{13}   &  3.66666666667 & -1.33 x_{1} & +  0.56 x_{11} & -12.00 x_{3} & -7.56 x_{4} & -4.78 x_{5} & -2.89 x_{6} & -1.67 x_{7} & + 10.44 x_{8}\\
\hline
z    &  -1.33333333333 & -2.33 x_{1} & -0.44 x_{11} & +  8.00 x_{3} & +  3.44 x_{4} & +  1.22 x_{5} & -5.89 x_{6} & +  6.33 x_{7} & -7.56 x_{8}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  2.66666666667 & +  4.67 x_{1} & +  0.89 x_{11} & +  2.00 x_{3} & +  0.11 x_{4} & -14.44 x_{5} & +  3.78 x_{6} & -8.67 x_{7} & -0.89 x_{8}\\
 x_{10}   &  1.0 & -6.00 x_{1} & +  0.33 x_{11} & +  4.00 x_{3} & -7.33 x_{4} & -2.67 x_{5} & -2.33 x_{6} & +  5.00 x_{7} & +  4.67 x_{8}\\
 x_{2}   &  0.333333333333 & -0.67 x_{1} & +  0.11 x_{11} & -1.00 x_{3} & -0.11 x_{4} & -0.56 x_{5} & +  0.22 x_{6} & -0.33 x_{7} & +  0.89 x_{8}\\
 x_{12}   &  1.33333333333 & + 12.33 x_{1} & -0.56 x_{11} &   & -3.44 x_{4} & +  1.78 x_{5} & -6.11 x_{6} & +  7.67 x_{7} & -11.44 x_{8}\\
 x_{13}   &  3.66666666667 & -1.33 x_{1} & +  0.56 x_{11} & -12.00 x_{3} & -7.56 x_{4} & -4.78 x_{5} & -2.89 x_{6} & -1.67 x_{7} & + 10.44 x_{8}\\
\hline
z    &  -1.33333333333 & -2.33 x_{1} & -0.44 x_{11} & +  8.00 x_{3} & +  3.44 x_{4} & +  1.22 x_{5} & -5.89 x_{6} & +  6.33 x_{7} & -7.56 x_{8}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  3.27777777778 & +  4.44 x_{1} & +  0.98 x_{11} & -0.17 x_{13} & -1.15 x_{4} & -15.24 x_{5} & +  3.30 x_{6} & -8.94 x_{7} & +  0.85 x_{8}\\
 x_{10}   &  2.22222222222 & -6.44 x_{1} & +  0.52 x_{11} & -0.33 x_{13} & -9.85 x_{4} & -4.26 x_{5} & -3.30 x_{6} & +  4.44 x_{7} & +  8.15 x_{8}\\
 x_{2}   &  0.0277777777778 & -0.56 x_{1} & +  0.06 x_{11} & +  0.08 x_{13} & +  0.52 x_{4} & -0.16 x_{5} & +  0.46 x_{6} & -0.19 x_{7} & +  0.02 x_{8}\\
 x_{12}   &  1.33333333333 & + 12.33 x_{1} & -0.56 x_{11} &   & -3.44 x_{4} & +  1.78 x_{5} & -6.11 x_{6} & +  7.67 x_{7} & -11.44 x_{8}\\
 x_{3}   &  0.305555555556 & -0.11 x_{1} & +  0.05 x_{11} & -0.08 x_{13} & -0.63 x_{4} & -0.40 x_{5} & -0.24 x_{6} & -0.14 x_{7} & +  0.87 x_{8}\\
\hline
z    &  1.11111111111 & -3.22 x_{1} & -0.07 x_{11} & -0.67 x_{13} & -1.59 x_{4} & -1.96 x_{5} & -7.81 x_{6} & +  5.22 x_{7} & -0.59 x_{8}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  2.0 & + 30.00 x_{1} & -2.00 x_{11} & -4.00 x_{13} & -25.00 x_{4} & -8.00 x_{5} & -18.00 x_{6} & + 46.00 x_{2} & +  0.00 x_{8}\\
 x_{10}   &  2.85714285714 & -19.14 x_{1} & +  2.00 x_{11} & +  1.57 x_{13} & +  2.00 x_{4} & -7.86 x_{5} & +  7.29 x_{6} & -22.86 x_{2} & +  8.57 x_{8}\\
 x_{7}   &  0.142857142857 & -2.86 x_{1} & +  0.33 x_{11} & +  0.43 x_{13} & +  2.67 x_{4} & -0.81 x_{5} & +  2.38 x_{6} & -5.14 x_{2} & +  0.10 x_{8}\\
 x_{12}   &  2.42857142857 & -9.57 x_{1} & +  2.00 x_{11} & +  3.29 x_{13} & + 17.00 x_{4} & -4.43 x_{5} & + 12.14 x_{6} & -39.43 x_{2} & -10.71 x_{8}\\
 x_{3}   &  0.285714285714 & +  0.29 x_{1} & +  0.00 x_{11} & -0.14 x_{13} & -1.00 x_{4} & -0.29 x_{5} & -0.57 x_{6} & +  0.71 x_{2} & +  0.86 x_{8}\\
\hline
z    &  1.85714285714 & -18.14 x_{1} & +  1.67 x_{11} & +  1.57 x_{13} & + 12.33 x_{4} & -6.19 x_{5} & +  4.62 x_{6} & -26.86 x_{2} & -0.10 x_{8}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.08 & +  1.20 x_{1} & -0.08 x_{11} & -0.16 x_{13} & -0.04 x_{9} & -0.32 x_{5} & -0.72 x_{6} & +  1.84 x_{2} & +  0.00 x_{8}\\
 x_{10}   &  3.01714285714 & -16.74 x_{1} & +  1.84 x_{11} & +  1.25 x_{13} & -0.08 x_{9} & -8.50 x_{5} & +  5.85 x_{6} & -19.18 x_{2} & +  8.57 x_{8}\\
 x_{7}   &  0.35619047619 & +  0.34 x_{1} & +  0.12 x_{11} & +  0.00 x_{13} & -0.11 x_{9} & -1.66 x_{5} & +  0.46 x_{6} & -0.24 x_{2} & +  0.10 x_{8}\\
 x_{12}   &  3.78857142857 & + 10.83 x_{1} & +  0.64 x_{11} & +  0.57 x_{13} & -0.68 x_{9} & -9.87 x_{5} & -0.10 x_{6} & -8.15 x_{2} & -10.71 x_{8}\\
 x_{3}   &  0.205714285714 & -0.91 x_{1} & +  0.08 x_{11} & +  0.02 x_{13} & +  0.04 x_{9} & +  0.03 x_{5} & +  0.15 x_{6} & -1.13 x_{2} & +  0.86 x_{8}\\
\hline
z    &  2.84380952381 & -3.34 x_{1} & +  0.68 x_{11} & -0.40 x_{13} & -0.49 x_{9} & -10.14 x_{5} & -4.26 x_{6} & -4.16 x_{2} & -0.10 x_{8}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{11}   &  1.0 & + 15.00 x_{1} & -12.50 x_{4} & -2.00 x_{13} & -0.50 x_{9} & -4.00 x_{5} & -9.00 x_{6} & + 23.00 x_{2} & +  0.00 x_{8}\\
 x_{10}   &  4.85714285714 & + 10.86 x_{1} & -23.00 x_{4} & -2.43 x_{13} & -1.00 x_{9} & -15.86 x_{5} & -10.71 x_{6} & + 23.14 x_{2} & +  8.57 x_{8}\\
 x_{7}   &  0.47619047619 & +  2.14 x_{1} & -1.50 x_{4} & -0.24 x_{13} & -0.17 x_{9} & -2.14 x_{5} & -0.62 x_{6} & +  2.52 x_{2} & +  0.10 x_{8}\\
 x_{12}   &  4.42857142857 & + 20.43 x_{1} & -8.00 x_{4} & -0.71 x_{13} & -1.00 x_{9} & -12.43 x_{5} & -5.86 x_{6} & +  6.57 x_{2} & -10.71 x_{8}\\
 x_{3}   &  0.285714285714 & +  0.29 x_{1} & -1.00 x_{4} & -0.14 x_{13} & -0.00 x_{9} & -0.29 x_{5} & -0.57 x_{6} & +  0.71 x_{2} & +  0.86 x_{8}\\
\hline
z    &  3.52380952381 & +  6.86 x_{1} & -8.50 x_{4} & -1.76 x_{13} & -0.83 x_{9} & -12.86 x_{5} & -10.38 x_{6} & + 11.48 x_{2} & -0.10 x_{8}\\
\end{array}\]


 $ x_{1} $ enters and Unbounded Dictionary!


 $ x_{1} $ enters and Unbounded Dictionary!
\end{document}
